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Given the plane curve, find tangent vector

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the plane curve [tex] \overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j}[/tex]
    Find the following when t= ∏/2
    [tex] Part A: \hat{T}(t)[/tex]
    [tex] Part B: \hat{B}(t)[/tex]
    [tex] Part C: \hat{N}(t)[/tex]
    2. Relevant equations

    [tex] \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}[/tex]
    [tex] \hat{T}(t)=\frac{\overrightarrow{r'(t)}}{|| \overrightarrow{r'(t)}||}[/tex]

    [tex]\hat{B(t)}=\frac{\overrightarrow{r'(t)\times r''(t) }}{||\overrightarrow{r'(t)\times r''(t)}||}[/tex]

    3. The attempt at a solution
    Part A
    [tex]\overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j} [/tex]

    [tex] \overrightarrow{r'(t)}=e^t[(cos(t)-sin(t))\hat{i} \:+\:(sin(t)+cos(t))\hat{j}] [/tex]

    [tex] \overrightarrow{r'(\frac{\pi }{2})}=e^\frac{\pi }{2}[(cos(\frac{\pi }{2})-sin(\frac{\pi }{2}))\hat{i} \:+\:(sin(\frac{\pi }{2})+cos(\frac{\pi }{2}))\hat{j}] [/tex]

    [tex] \overrightarrow{r'(\frac{\pi }{2})}=-e^\frac{\pi }{2}\hat{i}\;+\;e^\frac{\pi }{2}\hat{j} [/tex]


    [tex]\hat{T}(t) =\frac{-e^\frac{\pi }{2}\hat{i}\;+\;e^\frac{\pi }{2}\hat{j}}{ \sqrt{(-e^\frac{\pi }{2})^2\;+\;(e^\frac{\pi }{2})^2} }[/tex]

    Based off of Part A, plugging the numbers into Part B and C generate:
    [tex]\hat{B(t)}=\frac{\overrightarrow{r'(t)\times r''(t) }}{||\overrightarrow{r'(t)\times r''(t)}||}=0[/tex]
    [tex] \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}=0[/tex]

    Not sure if I solved this correctly.

    Any help would be great. Thank you.
     
  2. jcsd
  3. Feb 25, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You got T(t) right. That's about all you've really shown. Your formula for N(t) isn't even right. It's supposed to have a derivative in it.
     
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